BPS monopole in the space of boundary conditions

Abstract

The space of all possible boundary conditions that respect the self-adjointness of the Hamiltonian operator is known to be given by the group manifold U(2) in one-dimensional quantum mechanics. In this paper we study non-Abelian Berry’s connections in the space of boundary conditions in a simple quantum mechanical system. We consider a system for a free spinless particle on a circle with two point-like interactions described by the U(2) × U(2) family of boundary conditions. We show that, for a certain SU(2) ⊂ U(2) × U(2) subfamily of boundary conditions, all the energy levels become doubly-degenerate thanks to the so-called higher-derivative supersymmetry, and the non-Abelian Berry’s connection in the ground-state sector is given by the Bogomolny–Prasad–Sommerfield (BPS) monopole of SU(2) Yang–Mills–Higgs theory. We also show that, in the ground-state sector of this quantum mechanical model, the matrix elements of the position operator give the adjoint Higgs field that satisfies the BPS equation. It is also discussed that Berry’s connections in the excited-state sectors are given by non-BPS ’t Hooft–Polyakov monopoles.